Modélisation de l'écoulement diphasique dans les injecteurs Diesel

Cavitation modellingUp-to-date spray models donot take in account the non-stationary conditionsat the nozzle exit, because of the lack of knowledgeof nozzle exit conditions. As experimental investigationsare hardly possible in the injectors, themodelling tool is the solution to know those conditions.Several two-phase model types exist, dependingon flow conditions. Floryan and Rasmussen [43]have made a very good review of the existing models.We have seen the main characteristics of theflow, allowing us to choose the appropriate methods.We have classified the models in two families :the interface tracking and capturing methods, andthe others, i.e. methods that do not take in accountthe interface location.Interface is explicitely defined by the model• For interface tracking [44], interface is explicitelydefined and advected by the flow. The interface isdefined by markers which form a moving boundary,and an interpolation between these points ismade to represent the physical boundary. At eachtime step, the information concerning the markers(their location, their sequence) is stored. The advantageof this method is the access to sub-grid scalestructures. But this needs a big storage capacity.The main drawback is the points repartition on theboundary, i.e. several segments can be representedwith a lot of points, whereas others can be verybadly represented. Furthermore, in case of coalescenceor fragmentation, markers must be added orremoved. The best way to manage this has not beenfound yet.• Front capturing methods consist in definingexplicitely the interface by a volumic function. Theinterface is not represented but re-built. The mostused is the Volume of Fluid (VOF) method [45, 46,47, 48, 49, 50]. The principle is to represent for eachcell the volumic fraction by a function c(t,x) whichis singular at the interface :c(t,x) = 1 for the phase 1,c(t,x) = 0 for the phase 2.(2)In this method, a set of Navier-Stokes equationsis solved while the density of the fluid varies fromliquid to vapour densities, as following :ρ(t,x) = c(t,x)ρ 1 + (1 − c(t,x))ρ 2 (3)The vapour and the liquid are considered as incompressible.As we have seen, the interface is re-builtgeometrically. To keep a good accuracy, a lot ofcells are needed to determine the interface location.Furthermore, Topological changes like coalescenceor fragmentation are hardly performed, exceptwith a very good spatial resolution. Typically,10 cells must be assigned to define a bubble diameter[51]. Considering small bubbles whose diameteris approximately 5µm, the 3D computational gridcontains almost 200 million cells for a typical-sizedsingle-holed nozzle. As we have seen that the flowis tridimensional and that we have to compute inthe whole injector, the grid will reach more than 1billion cells for a five-holed injector.• By the same way, the Level Set method [52, 44]consists in defining a continuous and “sufficiently”regular function which is defined as :Φ (t,x) > 0 for the phase 1,Φ (t,x) < 0 for the phase 2,Φ (t,x) = 0 at the interface.The mixture density is given by :(4)ρ(Φ ) = ρ g + (ρ l − ρ g )H(Φ ) , (5)with H, Heaviside function, defined as following :⎧⎨ 0 if Φ < 01H(Φ ) =⎩2 if Φ = 01 if Φ > 0(6)Unverdi and Tryggvason [44] have used this methodfor 2D and 3D bubble foam advection. A methodhas been proposed by Smereka [52], in order tosmooth the Φ discontinuity at the interface. As amatter of fact, this method is hardly usable in caseof great density gradients, because of the front diffusion.• The bubble tracking method [51] is an attemptto link the so-called interface tracking models(which may be used for inclusions bigger than thecomputational cells) to the two-fluid models, whichare well adapted to inclusions far smaller than themesh cells. The model is a Eulerian-Lagrangianmodel, considering liquid as continuous phase. Themajor concept of the method is the ratio d ∗ = d∆x (dis the bubble diameter and ∆ x the computationalcell size). The aim of the model is to solve flowdynamics for 0 < d ∗ < 2 and d ∗ > 2. The massand momentum equations are solved for each phase,considering no mass transfer. For the continuousliquid phase, slip between the phases, transverselift force and interfacial drag force are not takenin account for d ∗ > 2. The equations for the dispersedvapour phase are written for each bubble,and the bubble shape is determined via graphicalcorrelations depending on three dimensionless numbers(Eötvös, Morton and Reynolds numbers) [53].The bubble geometry is used to calculate the voidfraction in each computational cell. This methodis clearly not adapted to Diesel engine nozzle flowmodelling, because of the bubble deformation dueto the shear stress. Moreover, the equations usedfor each vapour entity are likely to lead to a prohibitivecomputer time. Nevertheless, this method